Combine the strawberries, the balsamic vinegar, and 1 tablespoon of the brown sugar in a medium bowl. Let sit at room temperature while you make the biscuits.

Combine the flour, 2 tablespoons of the brown sugar, the baking powder, the salt, and the baking soda in a medium bowl. Add the butter and blend with a pastry cutter or your fingers until the mixture resembles coarse crumbs. Add ½ cup plus 2 tablespoons of the yogurt and ½ teaspoon of the vanilla extract and stir just until combined, then transfer the dough to a floured surface and knead 5 or 6 times.

With a rolling pin, roll out the dough until it’s ¾ inch thick (or pat it flat with your hands). Cut into rounds with a 2-inch biscuit cutter or glass and transfer to an ungreased baking sheet with the edges of the rounds touching. Gather up the dough scraps, roll them out, and repeat. Brush the biscuits with the milk, then sprinkle the granulated sugar over them. Bake until the biscuits are golden brown, 12 to 15 minutes.

While the biscuits are cooling, beat the cream with the whisk attachment of a stand mixer (or a handheld electric mixer) until soft peaks form, then beat in the remaining ½ cup yogurt, 2 tablespoons brown sugar, and 1 teaspoon vanilla extract. To serve, split each biscuit horizontally and top with a few tablespoons each of the strawberry mixture and the cream mixture.